F-invariant and E-invariant
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913727253577728 |
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| author | Cao, Peigen |
| author_facet | Cao, Peigen |
| contents | $F$-invariant for a pair of good elements (e.g. cluster monomials) in cluster algebras is introduced by the author in a previous work. A key feature of $F$-invariant is that it is a coordinate-free invariant, that is, it is mutation invariant under the initial seed mutations. $E$-invariant for a pair of decorated representations of quivers with potentials is introduced by Derksen, Weyman and Zelevinsky, which is also a coordinate-free invariant. The strategies used to show the mutation-invariance of $F$-invariant and $E$-invariant are totally different.
In this paper, we give a new proof of the mutation-invariance of $F$-invariant following the strategy used by Derksen, Weyman and Zelevinsky. As a result, we prove that $F$-invariant coincides with $E$-invariant on cluster monomials. We also give a proof of Reading's conjecture, which says that the non-compatible cluster variables in cluster algebras can be separated by the sign-coherence of $g$-vectors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06605 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | F-invariant and E-invariant Cao, Peigen Representation Theory Rings and Algebras 13F60, 16G10 $F$-invariant for a pair of good elements (e.g. cluster monomials) in cluster algebras is introduced by the author in a previous work. A key feature of $F$-invariant is that it is a coordinate-free invariant, that is, it is mutation invariant under the initial seed mutations. $E$-invariant for a pair of decorated representations of quivers with potentials is introduced by Derksen, Weyman and Zelevinsky, which is also a coordinate-free invariant. The strategies used to show the mutation-invariance of $F$-invariant and $E$-invariant are totally different. In this paper, we give a new proof of the mutation-invariance of $F$-invariant following the strategy used by Derksen, Weyman and Zelevinsky. As a result, we prove that $F$-invariant coincides with $E$-invariant on cluster monomials. We also give a proof of Reading's conjecture, which says that the non-compatible cluster variables in cluster algebras can be separated by the sign-coherence of $g$-vectors. |
| title | F-invariant and E-invariant |
| topic | Representation Theory Rings and Algebras 13F60, 16G10 |
| url | https://arxiv.org/abs/2503.06605 |